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Gridap.Adaptivity

The adaptivity module provides a framework to work with adapted (refined/coarsened/mixed) meshes.

It provides

  • A generic interface to represent adapted meshes and a set of tools to work with Finite Element spaces defined on them. In particular, moving CellFields between parent and child meshes.
  • Particular implementations for conformally refining/coarsening 2D/3D meshes using several well-known strategies. In particular, Red-Green refinement and longest-edge bisection.

Interface

The following types are defined in the module:

Gridap.Adaptivity.RefinementRule — Type

Structure representing the map between a single parent cell and its children.

Contains:

  • T :: RefinementRuleType, indicating the refinement method.
  • poly :: Polytope, representing the geometry of the parent cell.
  • ref_grid :: DiscreteModel defined on poly, giving the parent-to-children cell map.
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Gridap.Adaptivity.AdaptivityGlue — Type

Glue containing the map between two nested triangulations. The contained datastructures will depend on the type of glue. There are two types of AdaptivityGlue:

  • RefinementGlue :: All cells in the new mesh are children of cells in the old mesh. I.e given a new cell, it is possible to find a single old cell containing it (the new cell might be exactly the old cell if no refinement).
  • MixedGlue :: Some cells in the new mesh are children of cells in the old mesh, while others are parents of cells in the old mesh.

Contains:

  • n2o_faces_map :: Given a new face gid, returns
    • if fine, the gid of the old face containing it.
    • if coarse, the gids of its children (in child order)
  • n2o_cell_to_child_id :: Given a new cell gid, returns
    • if fine, the local child id within the (old) coarse cell containing it.
    • if coarse, a list of local child ids of the (old) cells containing it.
  • refinement_rules :: Array conatining the RefinementRule used for each coarse cell.
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Gridap.Adaptivity.AdaptedDiscreteModel — Type

DiscreteModel created by refining/coarsening another DiscreteModel.

The refinement/coarsening hierarchy can be traced backwards by following the parent pointer chain. This allows the transfer of dofs between FESpaces defined on this model and its ancestors.

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Gridap.Adaptivity.AdaptedTriangulation — Type

Triangulation produced from an AdaptedDiscreteModel.

Contains:

  • adapted_model :: AdaptedDiscreteModel for the triangulation.
  • trian :: Triangulation extracted from the background model, i.e get_model(adapted_model).
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The high-level interface is provided by the following methods:

Gridap.Adaptivity.refine — Function

function refine(model::DiscreteModel,args...;kwargs...) :: AdaptedDiscreteModel

Returns an AdaptedDiscreteModel that is the result of refining the given DiscreteModel.

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Gridap.Adaptivity.coarsen — Function

function coarsen(model::DiscreteModel,args...;kwargs...) :: AdaptedDiscreteModel

Returns an AdaptedDiscreteModel that is the result of coarsening the given DiscreteModel.

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Gridap.Adaptivity.adapt — Function

function adapt(model::DiscreteModel,args...;kwargs...) :: AdaptedDiscreteModel

Returns an AdaptedDiscreteModel that is the result of adapting (mixed coarsening and refining) the given DiscreteModel.

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Edge-Based refinement

The module provides a refine method for UnstructuredDiscreteModel. The method takes a string refinement_method that determines the refinement strategy to be used. The following strategies are available:

  • "red_green" :: Red-Green refinement, default.
  • "nvb" :: Longest-edge bisection (only for meshes of TRIangles)

Additionally, the method takes a kwarg cells_to_refine that determines which cells will be refined. Possible input types are:

  • Nothing :: All cells get refined.
  • AbstractArray{<:Bool} of size num_cells(model) :: Only cells such that cells_to_refine[iC] == true get refined.
  • AbstractArray{<:Integer} :: Cells for which gid ∈ cells_to_refine get refined

The algorithms try to respect the cells_to_refine input as much as possible, but some additional cells might get refined in order to guarantee that the mesh remains conforming.

  function refine(model::UnstructuredDiscreteModel;refinement_method="red_green",kwargs...)
    [...]
  end

CartesianDiscreteModel refining

The module provides a refine method for CartesianDiscreteModel. The method takes a Tuple of size Dc (the dimension of the model cells) that will determine how many times cells will be refined in each direction. For example, for a 2D model, refine(model,(2,3)) will refine each QUAD cell into a 2x3 grid of cells.

  function refine(model::CartesianDiscreteModel{Dc}, cell_partition::Tuple) where Dc
    [...]
  end

Notes for users

Most of the tools provided by this module are showcased in the tests of the module itself, as well as the following tutorial (coming soon).

However, we want to stress a couple of key performance-critical points:

  • The refining/coarsening routines are not optimized for performance. In particular, they are not parallelized. If you require an optimized/parallel implementation, please consider leveraging specialised meshing libraries. For instance, we provide an implementation of refine/coarsen using P4est in the GridapP4est.jl library.

  • Although the toolbox allows you to evaluate CellFields defined on both fine/coarse meshes on their parent/children mesh, both directions of evaluation are not equivalent. As a user, you should always try to evaluate/integrate on the finest mesh for maximal performance. Evaluating a fine CellField on a coarse mesh relies on local tree searches, and is therefore a very expensive operation that should be avoided whenever possible.

Notes for developers

RefinementRule API

Given a RefinementRule, the library provides a set of methods to compute the mappings between parent (coarse) face ids and child (fine) face ids (and vice-versa). The ids are local to the RefinementRule.

Gridap.Adaptivity.get_d_to_face_to_child_faces — Function

Given a RefinementRule, returns for each parent/coarse face the child/fine faces of the same dimension that it contains. Therefore, only fine faces at the coarse cell boundary are listed in the returned structure.

Returns: [Face dimension][Coarse Face id] -> [Fine faces]

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Gridap.Adaptivity.get_d_to_face_to_parent_face — Function

Given a RefinementRule, returns for each fine/child face the parent/coarse face containing it. The parent face can have higher dimension.

Returns the tuple (A,B) with

  • A = [Face dimension][Fine Face id] -> [Parent Face]
  • B = [Face dimension][Fine Face id] -> [Parent Face Dimension]
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Gridap.Adaptivity.get_face_subface_ldof_to_cell_ldof — Function

Given a RefinementRule of dimension Dc and a Dc-Tuple fine_orders of approximation orders, returns a map between the fine nodal dofs of order fine_orders in the reference grid and the coarse nodal dofs of order 2⋅fine_orders in the coarse parent cell.

The result is given for each coarse/parent face of dimension D as a list of the corresponding fine dof lids, i.e

  • [coarse face][coarse dof lid] -> fine dof lid
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AdaptivityGlue API

Gridap.Adaptivity.get_d_to_fface_to_cface — Function

For each child/fine face, returns the parent/coarse face containing it. The parent face might have higher dimension.

Returns two arrays:

  • [dimension][fine face gid] -> coarse parent face gid
  • [dimension][fine face gid] -> coarse parent face dimension
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New-to-old field evaluations

When a cell is refined, we need to be able to evaluate the fields defined on the children cells on the parent cell. To do so, we bundle the fields defined on the children cells into a new type of Field called FineToCoarseField. When evaluated on a Point, a FineToCoarseField will select the child cell that contains the Point and evaluate the mapped point on the corresponding child field.

Gridap.Adaptivity.FineToCoarseField — Type

Given a domain and a non-overlapping refined cover, a FineToCoarseField is a Field defined in the domain and constructed by a set of fields defined on the subparts of the covering partition. The refined cover is represented by a RefinementRule.

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